
theorem
  for A,B be real-membered set holds for y being Real holds y in B ++ A iff
    ex x,z being Real st x in B & z in A & y = x + z
  proof
    let A,B be real-membered set;
    let y be Real;
    hereby assume y in B++A; then
      consider x,w being Complex such that
A1:   y = x+w & x in B & w in A;
      reconsider x,w as Real by A1;
      take x,w;
      thus x in B & w in A & y = x + w by A1;
    end;
    given w,z being Real such that
A2: w in B & z in A & y = w + z;
    thus thesis by A2;
  end;
